scholarly journals Tietze Extension Theorem for n-dimensional Spaces

2014 ◽  
Vol 22 (1) ◽  
pp. 11-19 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Compact Subset ◽  
Closed Subset ◽  
Extension Theorem ◽  
Convex Compact ◽  
Arbitrary Subset ◽  
Empty Interior ◽  
Convex Compact Subset ◽  
Arbitrary Convex

Summary In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of En with a non-empty interior. This article is based on [20]; [23] and [22] can also serve as reference books.

scholarly journals Brouwer Fixed Point Theorem in the General Case

2011 ◽  
Vol 19 (3) ◽  
pp. 151-153 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Compact Subset ◽  
Point Theorem ◽  
Convex Compact ◽  
Empty Interior ◽  
Brouwer Fixed Point Theorem ◽  
Convex Compact Subset ◽  
Arbitrary Convex

Brouwer Fixed Point Theorem in the General Case In this article we prove the Brouwer fixed point theorem for an arbitrary convex compact subset of εn with a non empty interior. This article is based on [15].

scholarly journals An estimate for the total mean curvature in negatively curved spaces

2002 ◽  
Vol 66 (2) ◽  
pp. 267-273
Author(s):  
Albert Borbély
Keyword(s):  
Compact Subset ◽  
Mean Curvature ◽  
Smooth Boundary ◽  
Convex Compact ◽  
Empty Interior ◽  
Connected Manifold ◽  
Total Mean Curvature ◽  
Negatively Curved ◽  
Nonpositively Curved ◽  
Simply Connected

Let Mn be a nonpositively curved complete simply connected manifold and D ⊂ Mn be a convex compact subset with non-empty interior and smooth boundary. It is shown that the total mean curvature ∂D can be estimated in terms of volume and curvature bound.

scholarly journals Smoothness in spaces of compact operators

1988 ◽  
Vol 37 (2) ◽  
pp. 221-225 ◽  
Author(s):  
A. Sersouri
Keyword(s):  
Banach Spaces ◽  
Compact Subset ◽  
Compact Operators ◽  
Convex Compact ◽  
Convex Compact Subset

We prove that if X and Y are two (real) Banach spaces such that dim X ≥ 2 and dim Y ≥ 2, then the space K(X, Y) contains a convex compact subset C with dim C ≥ 2 (in the affine sense) which fails to be an intersection of balls. This improves two results of Ruess and Stegall.

scholarly journals A fixed point theorem for nonexpansive compact self-mapping

2014 ◽  
Vol 68 (1) ◽  
Author(s):  
T. D. Narang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Topological Space ◽  
Fixed Points ◽  
Compact Subset ◽  
Metric Space ◽  
Closed Subset ◽  
Continuous Family ◽  
Underlying Space ◽  
The Subject

AbstractA mapping T from a topological space X to a topological space Y is said to be compact if T(X) is contained in a compact subset of Y . The aim of this paper is to prove the existence of fixed points of a nonexpansive compact self-mapping defined on a closed subset having a contractive jointly continuous family when the underlying space is a metric space. The proved result generalizes and extends several known results on the subject

A biregular extension theorem from a fractal surface in

2011 ◽  
Vol 18 (1) ◽  
pp. 21-29
Author(s):  
Ricardo Abreu Blaya ◽  
Juan Bory Reyes ◽  
Tania Moreno García
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Extension Theorem ◽  
Fractal Surface ◽  
Fractal Boundary ◽  
Hölder Continuous

Abstract The aim of this paper is to prove the characterization on a bounded domain of with fractal boundary and a Hölder continuous function on the boundary guaranteeing the biregular extendability of the later function throughout the domain.

On Certain Onto Maps

1962 ◽  
Vol 14 ◽  
pp. 461-466 ◽  
Author(s):  
Isaac Namioka
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Dimensional Space ◽  
Extension Theorem ◽  
Normal Space ◽  
Continuous Map ◽  
Image Position

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined byA subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such thatand the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.

More on Extending Continuous Pseudometrics

1970 ◽  
Vol 22 (5) ◽  
pp. 984-993 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closed Subset ◽  
Locally Finite ◽  
Whole Space

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

Monomorphisms of Semigroups of Local Dendrites

1986 ◽  
Vol 38 (4) ◽  
pp. 769-780 ◽  
Author(s):  
K. D. Magill
Keyword(s):  
Continuous Function ◽  
Topological Space

When we speak of the semigroup of a topological space X, we mean S(X) the semigroup of all continuous self maps of X. Let h be a homeomorphism from a topological space X onto a topological space Y. It is immediate that the mapping which sends f ∊ S(X) into h º f º h−1 is an isomorphism from the semigroup of X onto the semigroup of Y. More generally, let h be a continuous function from X into Y and k a continuous function from Y into X such that k º h is the identity map on X. One easily verifies that the mapping which sends f into h º f º k is a monomorphism from S(X) into S(Y). Now for “most” spaces X and Y, every isomorphism from S(X) onto S(Y) is induced by a homeomorphism from X onto Y. Indeed, a number of the early papers dealing with S(X) were devoted to establishing this fact.

On the Edelstein Contractive Mapping Theorem

1975 ◽  
Vol 18 (5) ◽  
pp. 675-678
Author(s):  
Ludvik Janos
Keyword(s):  
Topological Space ◽  
Compact Subset ◽  
Contractive Mapping ◽  
Banach Contraction Principle ◽  
Contraction Principle ◽  
Banach Contraction ◽  
Point Set ◽  
The Banach Contraction Principle

AbstractLet X be a metrizable topological space and f:X→X a continuous selfmapping such that for every x ∈ X the sequence of iterates {fn(x)} converges. It is proved that under these conditions the following two statements are equivalent:1. There is a metrization of X relative to which f is contractive in the sense of Edelstein.2. For any nonempty f-invariant compact subset Y of X the intersection of all iterates fn(Y) is a one-point set. The relation between this type of contractivity and the Banach contraction principle is also discussed.

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